X-Box is likely to win, Nintendo will retain the number two spot and it is possible that Sony will survive(though I wouldn't say it is certain).
The first,
major, problem that Sony has its' absolutely horribly design. Whoever came up with the PS2 should be blind folded and hung up in the corporate offices for pinatas

Ask any developer working on the system, Sony made the same moronic mistake Nintendo did with the N64, made it too hard to develop for.
With Sony's main focus on teenage boys, a group that tends to be exceedingly unloyal(as proven by the fact that the PSX "won" the last round), they are nearly certain to fail the market Nintendo already has down(loyal customer base). Twenty to thirty million GameCubes are going to sell, that can be banked on. Sony, with their enormous production woes and launch that can be considered nothing but a completely failure(look at the installed base currently) outside of hollow hype, Sony has truly blown it. They missed the Cristmas season where they would have been the most powerful console on the market in the largest single market in the world(US is the largest, just not the highest per capita). Another disturbing trend for Sony is that certain titles are being cancelled for it already, not a very good show of faith.
Microsoft has a *very* large library of games lined up for launch. One of their main focuses was addressing developers desires, making the console, by comparison, extremely simplistic to make games for. So simple, in fact, that the schedules are running well ahead in most development houses. Many titles that were initially looking to be Q1 or Q2 '02 are now looking to be ready for launch with many launch titles gaining additional development time for fine tuning before they launch. Combine this with the already impressive list of developers for the X-Box and it is quite likely that the quality of launch titles will humiliate what Sony had very, very badly.
Nintendo is going to make games for the GameCube, they could likely sell ten to fifteen million consoles on that fact alone. They learned the lesson from their fvck up with the N64 and have made the GC nearly as easy as the X-Box to work with. Developers who have used all three have made a very clear line, the GC and X-Box are pleasures working with while the PS2 is the most difficult they have used.
Increased development difficulty increases production time, which increases costs to publishers and makes it far more difficult to hit launch dates.
Add to this the incredible performance rift between the GC and X-Box and Sony is nearly certain to fail, the big question in my mind is will this be their "Saturn" or their "DreamCast"?
Between the big N and the big M, I think MS has the edge. They are actively targetting the widest range of consumers and people do not already have preconcieved notions that they make "kids" games. Nintendo also is developing titles that are targetted at the more "Mature" crowd, but they will need to overcome their reputation before they make any real headway, something that I think they will accomplish to some extent but not to the level of MS.
MS also has a nine figure media blitz they are planning, making Sony's prior attempts look like a passing mention instead of massive hype machine. Expect MS to put the console in everyone's face, and with the level of power it has and some likely classics ready at launch(Halo is looking to have more gaming value then everyone of PS2's game to date, combined) the majority of people are going to be very impressed, unlike the situation with the PS2 looking noticably inferior to many Dreamcast titles when they are running side by side(think about this people, they only need to use HDTVs and run at 1080i res and they will make the PS2 look like sh!t, and they know it).
Then there is price. This one is likely to be a N win with MS matching wherever Sony is when they launch(though it is possible they will "go after" Nintendo's price point). If all three are ~$199 this upcoming holiday season, Sony may well pull a Sega by the end of '02 as far as hardware.